Fuzzy subset

When we have evaluated all the rules, an output variable might belong to two or more fuzzy subsets to different degrees. For example, in the enzyme problem one rule might conclude that the rate is low to a degree of 0.2 and another that the rate is low to a degree of 0.8. In aggregation, all the fuzzy values that have been calculated for each output variable are combined to provide a consensus value for the membership of the output variable in each... 

Functional Groups as Fuzzy Subsets of Molecular Electron Density... 

In the general scheme described in subsequent sections, a functional group is regarded as a fuzzy body of electronic charge cloud, a fuzzy subset of the electronic charge density cloud of the complete molecule. In this context, a functional group is a special case of a fuzzy fragment of a molecular body, obtained by some subdivision... 

For fuzzy sets used in this study various set operations are required. If A and B are fuzzy subsets of U, then the fuzzy intersection, that is, the result of the operation A and B is denoted by A i B, and is interpreted as a fuzzy subset C of set U, where the corresponding membership... 

A fuzzy generalization of the complement is obtained as the result of operation not A, denoted by A, interpreted as a fuzzy subset A of set... 

In some instances, operations between fuzzy subsets are also required if these subsets have different parent sets. If the operation A and B is applied to a fuzzy subset A of set U and a fuzzy subset B of a different set... 

As can be easily proven by a simple modification of the proof presented here for the unsealed fuzzy Hausdorff metric, this scaled fuzzy Hausdorff distance is also a metric in the space of fuzzy subsets of the underlying set X. A proof is given in subsequent text. 

Various fuzzy subsets B of 4 may have the fuzzy symmetry element B(/3) at different fuzzy levels /3 of the fuzzy Hausdorff-type similarity measure s. Here the concept of fuzzy subset is interpreted in the usual way for the fuzzy subset B of fuzzy set A the condition... 

A fuzzy set B is a maximal mass R-deficient subset of fuzzy set if B is an R-deficient fuzzy set, B c A, and if for all maximal R-deficient fuzzy subsets B of fuzzy set A, the relation m(B ) < m(B) holds. Whereas the fuzzy, maximal mass B-deficient subset B is not necessarily unique for a given fuzzy set A, nevertheless, the total mass m(B) is a unique number for each fuzzy set A. 

Consider two fuzzy sets A and B, of equal mass. Consider all rotated and translated versions A, and of sets A and B. Determine the maximal sealing factor fS( g, applied to a version A of A that allows the rescaled version scaled of to fit with B, where the statement u.scaied fitting within B is interpreted as the fuzzy subset relation... 

Consider a crisp or fuzzy subset A of the Euclidean space X, a (possibly approximate) symmetry element R, and the associated symmetry operator R. A fixed point of R is chosen as a reference point c e A, and a local Cartesian coordinate system of origin c is specified, with coordinate axes oriented according to the usual conventions with respect to the symmetry operator R, as described for crisp sets in Section XIII. 

Since the introduction of the MEDLA method, three more recent developments have extended the applications of the Mulliken-Mezey and the more general AFDF schemes. These developments, all utilizing various representations of additive fuzzy subsets of molecular electron densities, are the... 

A. Kaufmann, An Introduction to the Theory of Fuzzy Subsets, Vol. 1. Academic Press, San Diego, 1975. 

The term fuzzy set was first introduced by Zadeh (1965). This is the most adequate model for a description of nondefinitive situations which do not possess sharp boundaries. The mathematical model of the conflict situation, characterized in that the parameters of the empirical interaction potential model XA obtained on the basis of one set of properties do not correspond to the parameters XB obtained from another set of properties, may be represented as a set X of the alternative subsets of parameters with their fuzzy subsets, which map the unsharply formulated criteria i.e., as the system (X, f, fa,. . . , fR, L). In the framework of this problem, one needs to construct the following function in order to account for all possible criteria ... 

D = the fuzzy subset that results from selecting, for each X, the smallest membership value from any of the fuzzy subsets 5i through 5,2, under the assumption that the judgments of all subjects are equally rehable. 

B = (bI b2... h, b ) denotes subset of the evaluation set. bj reflects the position of jth evaluation in overall V. R is the fuzzy mapping from set Uto set V. According to the fuzzy mathematics theory, R would determine the fuzzy mapping. It makes a fuzzy subset Aoi U map to a fuzzy subset B of V. Triad U,V, R) constitutes a comprehensive evaluation model. 

The rows of i , indicate the membership degrees of fuzzy subset by single factor of first order indicators, To multiply row by the fuzzy weight vector ITj, the membership degree of each subset can be obtained, namely the result of the fuzzy comprehensive evaluation. 

A fuzzy mapping between two sets X and Y, T X Y can be defined in a similar way to that described in Section 5.3. Here, however, the mapping can quite easily be a many to many mapping or in other words the arrows of Fig. 5.3 may be such that more than one may start from a point in X, and more than one may arrive at a point in Y (Fig. 6.3). If A is a fuzzy subset of X and B is the... 

Now as we have noted earlier, fuzzy logic (FL) is an extension of MVL with truth values as fuzzy sets or more accurately as fuzzy restrictions on the truth. We will still call the truth space U defined on the interval [0, 1], but in FL a truth value will be a fuzzy subset t CU and Xr U- [0, 1]. We will still require rules to define the implication relation 1 as well as negation, conjunction C, disjunction D and equivalence. Zadeh suggested the use of the Lukasiewicz rules given above and they will be used in the rest of this book. Baldwin, Pilsworth and Guild [85, 86] have examined various alternative rules for implication. 

The range of possible values of a linguistic variable represents the variable s universe of discourse. For example, the universe of discourse of the linguistic variable completion date might have the range between 1 and 10 days, and include fuzzy subsets such as early, normal and late. 

Notice that (f2, e, P/) refers to a fuzzy probability space, denotes one of the fuzzy arithmetical operations, i.e., addition, subtraction, multiplication and division. It should also be noted that lx and Ox refer to the fuzzy subset of the real numbers 1 and 0, respectively. 

If X is a collection of objects, then the fuzzy subset a of X is defined as a set of ordered pairs ... 

The key concept behind the algorithm is the idea of the fuzzy partition of the input space Assuming a process with Ni input variables, the space of each input variable is evenly partitioned into a number of triangular fuzzy subsets. Then, fuzzy partitioning is extended to the entire input space so that a number of fuzzy subspaces are created, where each fuzzy subspace is defined as a combination of Ni particular fuzzy sets. 

Fuzzy Soft-Computing Methods and Their Applications Fuzzy Subset... 

If the elements of the universe are considered as variables then equation (9) reads as restriction if x = xq holds then y must be smaller than xq. Fuzzy relations are fuzzy subsets on X X Y determined by membership functions depending on two arguments of equal importance, e.g., fx x, y). In analogy to the former example we investigate the fuzzy relation R2. nearly smaller than . The membership function for Rj, I r, may be defined by... 

What is the difference between possibility and probability Zadeh writes Intuitively, possibility relates to our perception of the degree of feasibility or ease of attainment, whereas probability is associated with the degree of belief, likelihood, frequency, or proportion.According to Zadeh, the relation between the fuzzy proposition (statement) P containing the fuzzy variable X whose base set is U and the fuzzy set A that is a fuzzy subset of U is represented by... 

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